
If the free function is \(b(r)\)[1]}, [2]}, then from (REF ) we derive \(\mu (r)\)  and from (REF ), \(\mu (\beta ,\zeta )\) . If moreover \(b(r)\)  is invertible to get \(r(b)\) , then from (REF ), one gets \(\mu (r)=\mu (b)=\mu (\beta )\) . Then, the dynamical system reduces to a set of two equations for \(\beta ^{\prime }\)  and \(\theta ^{\prime }\) .

If the free function is \(\Phi (r)\)[3]}, [4]}, [5]}, we calculate \(\Phi ^{\prime }(r)\)  and then derive \(\theta (\beta ,\zeta )\)  and \(\frac{d\theta }{dl}(\beta ,\zeta )\)  with (REF ). We then deduce \(\mu (\beta ,\zeta )\)  from (REF ) with \(\theta (\beta ,\zeta )\)  a constraint between the three variables.

If the free function is \(\rho (r)\)[6]}, one gets \(\mu (\beta ,\zeta )\)  from (REF ).

If the free function is \(p(r)\)[6]}, [8]}, from (REF ) and (REF ) we derive the Tolman-Oppenheimer-Volkoff equation for static wormhole [9]}
\(\dot{p}=-\frac{(b+r^3 p) (p+\rho )}{2 r (r-b)}\) 
from which we get \(\mu (\beta ,\zeta )\)  with (REF ) and (REF ). If moreover \(r(P)\)  can be calculated, one derives \(\zeta (\beta ,\theta )\)  from (REF ) and thus \(\mu (\beta ,\theta )\)  from \(\mu (\beta ,\zeta )\) .

If the free function is \(w(r)\) , we get \(\mu (\beta ,\theta ,\zeta )=P(\beta ,\theta )/w(\zeta ,\beta )\)  using (REF ) and (REF ).

If the free function is \(w(z)\)  with \(z\)  the redshift and \(w\)  is invertible, and if one can calculate \(\rho (z)\) , then one gets \(\rho (w)=\rho (P/\mu )\) . Since \(r=r(\beta ,\zeta )\) , then from (REF ) it is possible to calculate a relation between \(\mu =\rho r^2\)  and a function of \((\mu ,\beta ,\theta ,\zeta )\) . This relation with the equations for \(\beta ^{\prime }\) , \(\theta ^{\prime }\)  and \(\zeta ^{\prime }\)  allow to study the phase space of static wormhole sustained by an equation of state initially given as a redshift function (see subsection REF ).

